Calculating Area of Plane Regions | r2sin(2theta) and r^2 < 4

[nanostudy]

Homework Statement

Find the area of the region in the plane where, r²sin(2theta) >2(sq.root3) , r^2 < 4

Homework Equations

The Attempt at a Solution

To try to visualize the problem a little better I converted from r²sin(2theta) to 2xy. However I'm confused after this, since I don't know what the upper limit to integrate is. Also, in the context of the question what does r^2 < 4 mean? Thanks very much. :)

 

[slider142]

Homework Statement

Find the area of the region in the plane where, r²sin(2theta) >2(sq.root3) , r^2 < 4

Homework Equations

The Attempt at a Solution

To try to visualize the problem a little better I converted from r²sin(2theta) to 2xy. However I'm confused after this, since I don't know what the upper limit to integrate is. Also, in the context of the question what does r^2 < 4 mean? Thanks very much. :)

Recall that $r = \sqrt{x^2 + y^2}$, so the inequality $r^2 < 4$ covers all points within a circle of radius 2 about the origin in the xy-plane. Sketch the graph of this bounding circle, and the graph of the bounding curve $2xy = 2\sqrt{3}$ (a rectangular hyperbola). Then shade in the regions that satisfy the inequality. Use the boundaries of those regions to define your integrals.

 

[HallsofIvy]

$r^2< 4$, in polar coordinates, is the same as "r< 2" since r is not negative. That is the interior of a circle, centered at the origin, with radius 2